Problem: Find the remainder when $2 \times 12 \times 22 \times 32 \times \ldots \times 72 \times 82 \times 92$ is divided by $5$.
Answer: We use the property $a \equiv b \pmod{m}$ implies $ac \equiv bc \pmod{m}$.

Since all numbers with the units digit of $2$ has a remainder of $2$ when divided by $5$ and we have $10$ numbers,  $$2 \times 12 \times 22 \times 32 \times \ldots \times 72 \times 82 \times 92 \equiv 2^{10} \equiv 1024 \equiv \boxed{4} \pmod{5}.$$